The thermodynamics of the Curie-Weiss model with random couplings.

We study the Curie-Weiss version of an Ising spin system with random, positively biased, couplings. In particular the case where the couplings ∈ij take the values one with probability p and zero with probability 1 - p which describes the Ising model on a random graph is considered. We prove that if p is allowed to decrease with the system size N in such a way that Np(N) ↑ ∞ as N ↑ ∞, then the free energy converges (after trivial rescaling) to that of the standard Curie Weiss model, almost surely. Equally, the induced measures on the mean magnetizations converge to those of the Curie-Weiss model. Generalizations of this result to a wide class of distributions are detailed.